Separable Differential Operators with Parameters

Abstract

In this paper, we study boundary value problems for parameter-dependent elliptic differential-operator equations with variable coefficients in smooth domains. Uniform regularity properties and Fredholmness of this problem are obtained in vector-valued $$ {\text{L}}_{p} $$ -spaces. We prove that the corresponding differential operator is positive and is a generator of an analytic semigroup. Then, via maximal regularity properties of the linear problem, the existence and uniqueness of the solution to the nonlinear elliptic problem is obtained. As an application, we establish maximal regularity properties of the Cauchy problem for abstract parabolic equations, Wentzell–Robin-type mixed problems for parabolic equations, and anisotropic elliptic equations with small parameters.